Some finiteness properties of the fundamental group of a smooth variety

C ^OMPOSITIO M ^ATHEMATICA

M ICHAEL P. A NDERSON

Some finiteness properties of the fundamental group of a smooth variety

Compositio Mathematica, tome 31, n

^o

3 (1975), p. 303-308

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303

SOME FINITENESS PROPERTIES OF

THE FUNDAMENTAL GROUP OF A SMOOTH VARIETY Michael P. Anderson

Noordhoff International Publishing

Printed in the Netherlands

In this _paper we prove that for _any smooth

variety

^{X over an}

algebraically

closed field of characteristic

p ~ 2, 3, 5

^the group

03A0(p)1(X)

is a

finitely presented pro-(p)-group.

^We recall that

03A0(p)1(X)

^denotes

the maximal

quotient

^of

03A01(X)

^{of order}

^prime

^{to p. In}

[8] ^Exposé

^II

this result is demonstrated for smooth X

provided

^{there exists a}

projective

^smooth

compactification

^{X of X such that}

XBX

^{is a divisor}

with normal

crossings

^{on X} and for all X

provided

^{we assume} strong resolution of

singularities

for all varieties of

dimension ~

^{n. Thus the}

result was

previously

^{known for X} of dimension ~ 2.

The essential new step ^{is Lemma 1} which allows us to reduce to the

case of dimension 2. The

proof

^{of this lemma uses}

Abhyankar’s

^work

on resolution of

singularities [1] together

^{with the}

technique

^of

fibering by

^{curves. We follow} the notation of

[7] Exposé

^{XIII and}

[8] Exposé

^II.

Let us now state our

proposition.

PROPOSITION 1: Let

X/k

be a connected smooth

variety

^{over the}

algebra- ically

^closed

field k of characteristic p ~ 2, 3, 5.

^Then

03A0(p)1(X) is a ^finitely

presented pro-(p)-group.

PROOF :

By [7] Exposé

^{IX it is} sufficient to prove the result for the elements of a Zariski

covering

^{of X. Thus the result follows}

by

^induction

on dimension from the result in dimension

2, [8] Exposé

^{II Theorem}

2.3.1,

^{and the}

following

^{lemma :}

LEMMA 1: Let X be a smooth

variety of dimension n ~

^{3 over the}

alge- braically

^closed

ground field k

^{and x a}

point ouf

^{x. Then x} has a Zariski

neighborhood

^U such that there exists an

algebraically

^{closed extension}

S2/k

and a smooth

variety V

^{over S2}

of dimension

^{n-1 and a}

morphism f : V ~ U

^such

that f ’

^{induces a}

surjection 03A01(V) ~ 03A01(U)

^{and an iso-}

morphism 03A0(p)1(V) ~ 03A0(p)1(U).

304

PROOF OF LEMMA 1: We

proceed by

^{induction on} the dimension of X.

Let U be an affine

neighborhood

^{of x.}

By [1]

Birational Resolution there exists a smooth

projective

model of the function field

k(U).

^{Let U}

be a

projective compactification

^{of U.}

By [1]

^Dominance there exists

a smooth

projective variety

^X’

together

^{with a} birational

morphism

X’ ~

U. By [1]

^Global Resolution there exists ^a smooth

projective variety

^X"

together

^{with a} birational

morphism

^{X" ~}

^LI

and such that the inverse

image

^of

0B U

^{is a} divisor with normal

crossings

^{on X". Let}

U" be the

complement

of this divisor. Then the _{map g :} U" ~ U is a

proper birational

mapping

^of smooth varieties. The

subvariety

^of

points

of U

where 9

^{is not an}

isomorphism

^{is of}

codimension ~

^{2. Thus}

by

^the

Purity

^Theorem

[7] Exposé X.3, g

^{induces an}

isomorphism

By [9], [5], or [10],

^a

general hyperplane

section of

U",

^{call it}

E gives

a smooth surface in U" such that

Thus the lemma is

proved

^{for n = 3.}

Now assume n &#x3E; 3.

By [4] Exposé XI,

^{x has a Zariski}

neighborhood

W which admits an

elementary fibration g :

^{W -} W’ with W’ smooth of dimension n -1.

Moreover, by [6] Proposition

^{2.8 we} may ^assume that g admits a finite etale multisection i.e. there exists a finite etale _map

s : S ~ W’

together

^{with a} closed immersion i : S ~ W such that

gi

^{= s.}

Let y

⁼

g(x). By induction y

^{admits a Zariski}

neighborhood

^{U’ in W’}

such that there exists a smooth

variety

^V’ of dimension n - 2 and a

morphism f’ :

^{V’ -} U’ such that

f’

^{induces an}

isomorphism

^{of the}

(p)-completions

of the fundamental _groups of V’and U’. Let U ⁼

g-1(U’)

and V

= V’ U,

^{U with}

projections f : V - U and g’ :

^{V - V’. Then}

g’

^{is an}

elementary

^fibration

admitting

^an etale multisection.

Letting

^C

be a

geometric

^fiber

of g’,

^we

have, by [7] ^Exposé

^XIII

Proposition 4.3,

exact sequences

Let K be the kernel of the

homomorphism 03A0’1(U) ~ 03A0’1(V)

^{and K’}

the kernel of

n1(V’) --+ 03A01(U’).

Then the natural map K - K’ is ^an

isomorphism. Moreover, by hypothesis

^K’ is contained in the closed normal

subgroup

^of

n1(V’) generated by

^the

Sylow

p

subgroups

^of

03A01(V’).

^{Since any}

^Sylow

^p

^subgroup

^of

03A01(V’)

^{is the}

^image

^{of a}

^Sylow

p

subgroup

^of

03A0’1(V),

^{K is} also contained in the

subgroup generated by

^the

conjugates

^{of the}

Sylow

p

subgroups.

^{Thus K} is contained in the kernel of

03A01(V) ~ 03A0(p)1(V).

Therefore the

homomorphism

is

injective and, by

^{the five}

lemma,

^{it is}

surjective.

^Thus the lemma and

proposition

^are

proved.

Using Proposition

^{1 and} standard descent

techniques

^{we can weaken}

the resolution

hypotheses required

^to prove finite

presentation

^of

03A0(p)1(X)

^for

^arbitrary

^{X. We shall say that a}

^point

^{x of a}

^variety

^{X admits}

a ’weak resolution

of singularities’

if there exists a Zariski

neighborhood

U of x in X and a

morphism

of effective descent for the category ^{of etale}

coverings f :

^{U’ ~ U such that U’ is a smooth}

variety.

^{We have then}

the

following :

PROPOSITION 2: Let X be a

variety

^{over an}

algebraically

^closed

field of

characteristic

p ~ 2, 3,

^{5. Assume that} every

point of

^{X admits a weak}

resolution

of singularities.

^Then

03A0(p)1(X) is a finitely presented pro-(p)-

group.

COROLLARY : Let X be a

variety of

^{dimension 3 over an}

algebraically

closed

field of

characteristic

p ~ 2, 3, 5.

^Then

03A0(p)1(X) is a ^finitely

^p

presented pro-(p)-group.

PROOF :

Proposition

^{2 is a}

straightforward application

^of

[7]

^IX.5

together

^with

Proposition

^{1. The}

Corollary

follows from

Proposition

²

and

Abhyankar’s

^{results on}

resolution [1].

As another

application

^{of the}

fibering by

^{curves method we will}

outline a

proof

^{of the}

following

^{result :}

PROPOSITION 3

(Kunneth Formula) :

^{Let X and Y} be connected varieties ^over the

algebraically

^closed

field k of

characteristic _p. Then the natural

homomorphism

306

is an

isomorphism.

In

[7] Exposé

^{XIII this}

proposition

^is demonstrated

using

^the

hypoth-

esis of strong resolution of

singularities.

^{We avoid the use} of resolution of

singularities

^{as follows :}

First we consider the case where X and Y are normal varieties. Then it is sufficient to prove the formula for some non-trivial _open subsets of X and Y Choose U in X and V in Y such that U and admit

elementary

fibrations

f :

^{U ~ U’}

and g :

^{V - V’} with etale multisections.

By

induction on the dimensions of U and V we may ^assume the

proposition

holds for U’ and V’. Let C and D be

geometric

^{fibers of}

f and g

respec-

tively.

^Since

f and g are elementary

fibrations

admitting

etale multi- sections we have the

following

^exact sequences

Arguing

^{now as in the}

proof of Lemma 1,

^{we see that the natural homo-}

morphism

induces an

isomorphism

^on

(p)-completions.

Consider now the case in which Y is assumed

normal,

^{and X is}

arbitrary.

Let X’ - X be the normalization of

X,

^{and define}

Let

X’03B1,

^{a E}

03A00(X’),

be the connected components ^{of X’. Then}

by [7]

^IX

Theorem

5.1, 03A01(X) is

^{the free}

^product

^{of the}

groups 03A01(X03B1)

^{and the}

free _group

generated by

^the elements of the set

flo(X")

^{modulo certain}

relations defined

by

^the

projections:

Thus the same

description

^{holds for}

nr)(X)

^after

^replacing

^{all the groups}

involved

by

^their

prime

to p

completions. Moreover,

^{the same result}

applies

^{to X’ x Y ~ X x 1: This}

gives

^a

description

^of

03A0(p)1(X ^Y)

^{as the}

free

product (in

^the

category of pro-(p)-groups)

^{of the} groups

03A0(p)1(X03B1

^x

^Y)

and the free

pro-(p)-group generated by

the elements of the set

no(X" x ^Y)

modulo relations defined

by

^the

projections :

It is

long

^and

tedious,

^but

straightforward,

^{to check}

that,

^since

the above relations force

Now

applying

^{the same} argument ^as above without the

assumption

that Y is normal

(which

is valid because we

just

verified that

for

Xa

^{normal and Y}

arbitrary) gives

the result for X and Y

arbitrary

varieties.

BIBLIOGRAPHY

[1] S. ABHYANKAR : Resolution of Singularities of Embedded Algebraic Surfaces. ^Academic Press, ^New York, ^1966.

[2] MICHAEL P. ANDERSON: Profinite Groups ^{and the} Topological Invariants of Algebraic

Varieties. Princeton Ph.D. thesis, ^1974.

[3] MICHAEL P. ANDERSON : EXACTNESS PROPERTIES OF PROFINITE COMPLETION FUNCTORS.

Topology ¹³ (1974) ^229-239.

[4] ^M. ARTIN, ^A. GROTHENDIECK, ^{J. L.} VERDIER: Theorie des Topos ^et Cohomologie

Etale des Schemas. Lecture Notes in Mathematics 305 (1973).

308

[5] ^{WOUT DE} BRUIN: Une forme algebrique du theoreme de Zariski pour 03A01. ^{C. R. Acad.}

Sci. Paris Ser. A-B 272 (1971) ^A769-A771.

[6] E. M. FRIEDLANDER: The Etale Homotopy Theory ^{of a} Geometric Fibration.

Manuscripta Mathematica 10 (1973) ^209-244.

[7] A. GROTHENDIECK et al.: Revetements Etales et Groupe Fondamental. Lecture Notes in Mathematics 224 (1971).

[8] A. GROTHENDIECK et al.: Groupes de Monodromie en Geometrie Algebrique.

Lecture Notes in Mathematics 288 (1972).

[9] HERBERT POPP: Ein Satz vom Lefschetzschen Typ ^{Uber die} Fundamentalgruppe quasi-projectiver Schemata. Math. Z. 116 (1970) ^143-152.

[10] M. RAYNAUD: Theoreme de Lefschetz en Cohomologie des Faisceux coherents et en cohomologie ^etale. Ann. E. N. S. 4 (1974) ^29-52.

(Oblatum 31-1-1975 &#x26; 26-VI-1975) Department of Mathematics Yale University

New Haven, Connecticut 06520 U.S.A.